A diagnostic tool includes a communications interface, a memory storing instructions, and at least one processor coupled to the communications interface and to the memory. The at least one processor is configured to execute the instructions to perform operations including: receiving measurement data of a response to a flow-mediated dilation (FMD) test; determining a value for an FMD parameter based on the received measurement data; establishing a diagnostic threshold based on patient medical information; determining a diagnostic result of the FMD test by comparing the FMD parameter value to the diagnostic threshold; and providing the diagnostic result of the FMD test through a digital interface of the diagnostic apparatus.
Legal claims defining the scope of protection, as filed with the USPTO.
. A diagnostic system comprising:
. The diagnostic system of, wherein the FMD parameter value comprises a patient-specific measurement resulting from the FMD test.
. The diagnostic system of, further comprising a data analysis tool configured to receive the image data and derive the measurement data from the received image data.
. The diagnostic system of, wherein the measurement tool comprises an ultrasound imaging device.
. The diagnostic system of, wherein the diagnostic threshold comprises a static threshold or a dynamic threshold.
. The diagnostic system of, wherein the patient information corresponds to at least one of age, family medical history, smoking history, and sex.
. The diagnostic system of, wherein the at least one processor is further configured to execute the instructions to perform operations comprising:
. The diagnostic system of, wherein the FMD test is a brachial artery flow-mediated dilation test.
. The diagnostic system of, wherein the diagnostic threshold comprises a range of thresholds corresponding to the patient medical information, and wherein providing the diagnostic result through the output module further comprises presenting a graphical representation of the range of thresholds and a marker denoting the diagnostic result, the marker being presented within the graphical representation in relation to the range of thresholds.
. The diagnostic system of, wherein the diagnostic result comprises a prediction in a physical status of a patient of the respective FMD test.
. The diagnostic system of, wherein the at least one processor is further configured to execute the instructions to perform operations comprising:
. A method comprising:
. The computer-implemented method of, wherein the FMD parameter value comprises a patient-specific measurement resulting from the FMD test.
. The computer-implemented method of, wherein the diagnostic threshold comprises a static threshold or a dynamic threshold.
. The computer-implemented method of, further comprising:
. The computer-implemented method of, further comprising applying at least one of a machine learning process or neural network processing to adjust one or more algorithms used to determine the FMD parameter value and the diagnostic threshold from the measurement data.
. A tangible, non-transitory computer-readable medium storing instructions that, when executed by at least one processor, cause the at least one processor to perform a method, comprising:
. The tangible, non-transitory computer-readable medium of, containing further stored instructions that, when executed by at least one processor, cause the at least one processor to further perform:
Complete technical specification and implementation details from the patent document.
This application claims benefit of U.S. Provisional Patent Application No. 63/116,420 filed Nov. 20, 2020, the entirety of which is incorporated by reference herein.
This invention was made with government support under NSF 1511096 awarded by the National Science Foundation. The government has certain rights in the invention.
The present disclosure relates to a diagnostic tool, and, more particularly, to a diagnostic tool for analyzing and using the results of a flow mediated dilation test.
One of the indicators of cardiovascular health is the structural integrity of arterial walls. One method for assessing the physical state of the arterial wall is the noninvasive and relatively inexpensive brachial artery Flow-Mediated Dilation (BAFMD) test, in which blood flow through the brachial artery is obstructed temporarily (for about five minutes) using a pressure cuff wrapped around the upper arm, causing the artery to become almost completely closed. The cuff is then suddenly deflated, allowing the flow to rush back in while the dilating artery is monitored until full recovery, using an ultrasound scanner. FMD metrics have been proposed in the past as potential cardiovascular health indicators. Correlations have been found linking abnormal FMD results with many underlying conditions and risk factors directly affecting cardiovascular health. Using high resolution ultrasound imaging, the impairment of BAFMD due to cigarette smoking has been investigated. The conclusions of the said study corroborate the findings of Celermajer, D. S. et al. Cigarette smoking is associated with dose-related and potentially reversible impairment of endothelium-dependent dilation in healthy young adults. Circulation 88, 2149-2155 (1993) reported on the influence that smoking has on the relationship between the arterial diameter and the blood velocity in the brachial artery. BAFMD's predictive power when it comes to the short-term development of early stage renal dysfunction has been established. In a study involving 38 obese men, visceral obesity was linked to BAFMD impairment. In several other studies, monitoring BAFMD has been shown to be instrumental for diagnosing cardiovascular diseases (CVDs) and assessing cardiovascular health. However, there is a significant lack of understanding of the fundamental biophysics governing the FMD process, which prevents it from being an effective and pervasive diagnostic for CVDs. The present disclosure includes a system and device that evaluate the output and results of an FMD process to produce a diagnostic tool that overcomes the previous issues of interpreting an FMD test.
The summary of the disclosure is given to aid understanding of flow mediated dilation, and more particularly, to a diagnostic tool for analyzing and using the results of a flow mediated dilation test, and not with an intent to limit the disclosure. The present disclosure is directed to a person of ordinary skill in the art. It should be understood that various aspects and features of the disclosure may advantageously be used separately in some instances, or in combination with other aspects and features of the disclosure in other instances. Accordingly, variations and modifications may be made to the systems, devices, and their methods of operation to achieve different effects. Certain aspects of the present disclosure provide a system, method, and non-transitory computer readable medium for flow mediated dilation testing and analysis.
In one or more embodiments, a diagnostic apparatus includes a communications interface, a memory storing instructions, and at least one processor coupled to the communications interface and to the memory. In one or more cases, the at least one processor is configured to execute the instructions to perform operations. In one or more cases, the operations include receiving measurement data of a response to a flow-mediated dilation (FMD) test. In one or more cases, the operations include determining a value for an FMD parameter based on the received measurement data. In one or more cases, the operations include establishing a diagnostic threshold based on patient medical information. In one or more cases, the operations include determining a diagnostic result of the FMD test by comparing the FMD parameter value to the diagnostic threshold. In one or more cases, the operations include providing the diagnostic result of the FMD test through a digital interface of the diagnostic apparatus.
In one or more embodiments, a computer-implemented method includes receiving image data of a patient from a medical diagnostic tool. In one or more cases, the image data corresponds to a response to a flow-mediated dilation (FMD) test. In one or more cases, the computer-implemented method includes deriving measurement data from the received image data. In one or more cases, the computer-implemented method includes determining a value for an FMD parameter based on the measurement data. In one or more cases, the computer-implemented method includes establishing a diagnostic threshold based on patient medical information. In one or more cases, the computer-implemented method includes determining a diagnostic result of the FMD test by comparing the FMD parameter value to the diagnostic threshold. In one or more cases, the computer-implemented method includes providing the diagnostic result of the FMD test through a digital interface of a diagnostic apparatus.
In one or more embodiments, a tangible, non-transitory computer-readable medium storing instructions that, when executed by at least one processor, cause the at least one processor to perform a method. In one or more cases, the method includes receiving image data of a patient from a medical diagnostic tool. In one or more cases, the image data corresponds to a response to a flow-mediated dilation (FMD) test. In one or more cases, the method includes deriving measurement data from the received image data. In one or more cases, the method includes determining a value for an FMD parameter based on the measurement data. In one or more cases, the method includes establishing a diagnostic threshold based on patient medical information. In one or more cases, the method includes determining a diagnostic result of the FMD test by comparing the FMD parameter value to the diagnostic threshold. In one or more cases, the method includes providing the diagnostic result of the FMD test through a digital interface of a diagnostic apparatus.
The present disclosure relates to a diagnostic tool for processing and analyzing results of a Flow-Mediated Dilation (FMD) test.
is a block diagram of an exemplary diagnostic system. The diagnostic systemmay include a diagnostic tool, a measurement tool, a data analysis tool, and a network. As described here, the diagnostic systemutilizes physics modeling of mechanical activity within a body of a patient. The diagnostic tool, measurement tool, and/or data analysis toolmay include computing devices configured to perform one or more steps of a process to provide a useful output for evaluating a patient based on measured and calculated data. The diagnostic toolmay be configured to receive measurement information from the measurement tooland feedback information from the data analysis tooland analyze the received information to produce a useful result helpful in evaluating the patient.
The measurement toolmay be a medical device configured to collect data from patient. For example, the measurement toolmay be an imaging device, such as an ultrasound imaging device. The measurement toolmay be configured to detect a patient response to an FMD process. The measurement tool, in some embodiments, may work in conjunction with the data analysis toolto provide useful information to the diagnostic tool. For example, the measurement toolmay provide images or imaging data to the data analysis tool.
The data analysis toolmay analyze the images to determine values for one or more FMD variables. The FMD variables may be indicators of a patient's health, as described herein. The diagnostic toolmay receive the FMD variables and perform an analysis to provide useful information about the patient's heath. In some embodiments, the data analysis toolmay be integrated into the diagnostic tool.
The networkmay be a communication device configured to connect the components of the diagnostic system, such as the diagnostic tool, the measurement tool, and/or the data analysis tool. The diagnostic tool, measurement tool, and data analysis toolmay each include a respective communications interface that are communicatively coupled to the network. In one or more cases, the communications interfaces may be configured to transmit and receive data to one or more other devices, such as, but not limited to, the diagnostic tool, measurement tool, and data analysis tool. The networkmay be a wireless or wired communication device. In an exemplary embodiment, the networkis a wireless connection, such as one or more of WiFi, Bluetooth®, or the Internet. In other embodiments, the networkmay include wired or integrated communication connections. The networkenables data communication between the components of the diagnostic system.
is a block diagram of an exemplary computing device. The computing devicemay include at least a processing unit, a memory unit, and one or more input/output device. The computing devicemay, in some embodiments, be connected to a database, such as a data repository. The computing devicemay be representative of one or more of the diagnostic tool, the measurement tool, or the data analysis tool.
In an exemplary embodiment, the computing deviceis an example of at least the diagnostic tooland may include in whole or in part the measurement tooland/or the data analysis tool. In some embodiments, the measurement tooland/or the data analysis toolare included via data connections at the input/output device(s).
The computing devicemay be a general or specialized computing system or component configured to receive information associated with an FMD process applied to a patient and provide output that is practically applicable to evaluating the health of the patient. In an exemplary embodiment, the computing deviceincludes a plurality of engines or modulesconfigured to perform processes for producing output based on measured input. The modulesmay include, in an exemplary embodiment, a modeling module, an analysis module, an output module, and a learning module. The computing devicemay include a digital user interface configured to display information, such as a diagnostic result of an FMD test, as described herein.
The modeling modulemay be configured to use measurement data to model an effect of an FMD process applied to a patient. The modeling of the FMD process is further described herein. The modeling modulemay be configured to use information from the data analysis toolto determine one or more parameters related to the FMD process for a particular patient. The parameters may include (and are sometimes referred to herein as) FMD variables that are patient-specific measurements occurring as a result of the FMD process. In one example, the modeling modulemay be configured to receive data from the data analysis toolon the basis of images received by the measurement tool. The modeling modulemay be configured to determine one or more parameters based on this data.
The analysis modulemay be configured to receive the parameters determined by the modeling moduleand identify an output to provide to a user of the diagnostic tool. For example, the analysis modulemay be configured to compare the one or more parameters to one or more thresholds to determine a diagnostically-useful result of the FMD process. For instance, if a particular parameter is higher than a given threshold, the analysis modulemay determine a particular risk factor associated therewith. The thresholds may be, for example, history, sex, etc. In some embodiments, the thresholds may be determined by the data analysis tool, such as based on patient-specific data input. For example, the data analysis toolmay retrieve a patient medical file and determine one or more thresholds to provide to the diagnostic tool. The various thresholds may be static or dynamic. For example, static thresholds may pertain to specific subgroups (e.g., categorized by age, ethnicity, gender, lifestyle, BMI, genetic predisposition, etc.) of healthy individuals. In some cases, the static thresholds may be determined based on controlled clinical studies, such that the modules provided herein, such as the analysis module, may evaluate the FMD parameters of healthy individuals in a given subgroup, and compare them to the parameter values corresponding to the patients belonging in the same subgroup, with various cardiovascular conditions and risk factors. In yet another example, as the modules allow for patient specific evaluations, a dynamic threshold may be determined based on moving averages of the parameter values obtained for a given individual over the course of several regular physical examinations that may include the BAFMD test. By determining the dynamic threshold based on moving averages, the modules may accommodate the given individual's normal aging process.
The output modulemay be configured to provide the results produced by the analysis moduleto a user. For example, the output modulemay be configured to provide one or more data points determined based on threshold comparisons to a digital user interface viewable by the user. In one example, the output modulemay be configured to output notifications associated with potential health risks determined based on the analysis module. In some embodiments, the output modulemay be configured to provide a graphical representation of the measured parameters and/or the applied thresholds. For example, the output modulemay provide a threshold range for a normal parameter and a marker denoting the measured parameter for the patient in relation to the threshold range. The output moduleprovides a tangible medium for enabling the diagnostic toolto practically apply the results of an FMD process to a diagnostic process. For example, the output moduleprovides a user with a quick and efficient result for diagnostic use based on an FMD process applied to a patient.
The learning moduleis a feedback component configured to work in conjunction with other components of the diagnostic systemto improve the underlying processes. For example, the learning modulemay be configured to provide data back to the data analysis tool, such as threshold comparison results or additional patient information input by the user. For example, a user may input actual health data for the patient that is indicative of patient health after the FMD process and analysis occurs. The learning modulemay use the additional information to tune algorithms and processes for determining FMD variables and thresholds. In some embodiments, the learning modulemay be configured to use machine learning and/or neural network processing to adjust the algorithms used to determine the FMD variables and/or thresholds for evaluating a patient's health.
The disclosed diagnostic system is related to the FMD process in which blood flow is cut off for a period of time in a patient's artery (e.g., brachial artery). The highly transient FMD process occurs at two main time scales, including that of a heartbeat (pulsation period) and that of the artery's dilation soon after the uncuffing process. Since arterial walls are not rigid, flow conditions will influence the artery's diameter change, which will in turn affect the flow conditions, making the problem fully coupled via a two-way fluid-structure interaction. The most challenging aspect of this system stems from the fact that, the arterial wall's mechanical properties are not constant throughout the FMD process, due to a physiological phenomenon known as the mechanotransduction. When the wall shear stress (WSS) changes, the endothelial cells (ECs) lining the arterial wall sense it, and through a complex network of biochemical signal pathways, instruct the artery's compliance to change accordingly. The microstructure through which ECs sense WSS, is the negatively charged Endothelial Glycocalyx Layer (EGL), a soft porous layer of proteoglycans and glycoproteins lining blood vessels' inner surface. In a set of in-vitro experiments, it has been shown that the EGL's structural configuration is indifferent to a disturbed flow lacking a forward component. Only when a forward shear stress was added during the FMD process, structural remodeling of the EGL could be observed. This preferential behavior towards forward (non-oscillatory) flows has also been observed when it comes to the ECs' shear-stress-induced release of the vasodilators responsible for increasing the compliance of arterial walls in response to an elevated WSS. A significant increase in vasodilation stimulation has been observed after a prolonged 24-hour exposure to a laminar flow. In another experimental study, it was shown that a steady laminar flow induced a nitric oxide (NO, a prominent vasodilator) synthesis that was dependent on the WSS magnitude (or step-change magnitude). On the other hand, upregulation of NO release failed when turbulent flow was introduced. On the flip side, reduced blood flow, as in congestive heart failure for example, flow mediated vasodilation is attenuated in-vivo. On the cellular level, mechanotransduction induced by WSS has been extensively studied via experimental investigations, analytical modeling, and numerical simulations. The EGL's physiological function as a mechanosensor and transducer has been well established. The transfer of fluid WSS at the EGL-fluid interface, through the matrix, to the form of solid stress in the endothelial cell's body has been described. Moreover, it has been shown that the EGL's presence is required for the ECs' responsiveness to WSS. Defective ECs have been shown to be unable to align themselves with a laminar flow even after a prolonged exposure. Atomic-scale molecular simulations have been used to discern the specific proteins acting as mechanosensors in the EGL. It was further established that the GPC1 core protein transmits the sensed shear stress to the EC's surface, resulting in NO production. Low shear stress has been shown to inhibit NO production, whereas high levels activate it. In the same study, when the aforementioned GPC1 core protein was removed, the effect that shear stress has on NO production was severely attenuated.
Collectively, the studies referenced above confirm the intimate dependence of the observed FMD response on the integrity of mechanotransduction taking place at the inner surface of the arterial wall, through sensing the highly transient changes in WSS levels after uncuffing. Guided by the knowledge acquired from all those studies probing mechanotransduction and vasodilator production on the microscopic level, the present disclosure relies at least in part on a feedback loop that macroscopically describes the mechanisms underlying the observed FMD response.is a depiction of the feedback loop. When the blood flow is abruptly allowed back through the shrunken artery, the drastic WSS increase is picked up by the endothelial cells, initiating signal pathways that will stimulate the change of the mechanical properties (i.e. increasing compliance due to vasodilation stimulation) of the arterial wall. The artery's diameter would then respond by increasing under the fluid's pressure, and hence initiating dilation and decreasing the flow speed along with the WSS, which completes the feedback loop that makes FMD a self-modulating process.
Based on this, a physics-based model driven by the feedback loop depicted inhas been developed to describe the FMD response. This model was tested and observed in 5 healthy human subjects. The model correctly predicted a key feature in the beginning of the response that was experimentally observed across all subjects, and that conventional viscoelastic models fail to explain. Dimensionless parameters, each with a clear physical meaning, arose from the model. These dimensionless parameters can be used by the disclosed diagnostic system.
The evaluation of these parameters for each subject based on their FMD response, provided a quantitative description of the physical state of their artery. This evaluation links the microscopic underpinnings of endothelial mechanotransduction, to macroscopic observable, measurable, and physically meaningful quantities. While previous studies included some assumptions about arterial wall thickness, the present disclosure includes a diagnostic tool that utilizes a physics-based model describing the BAFMD response, in which the thickness of the arterial wall is not neglected. Mechanotransduction is accommodated by introducing a conceptual property, visualized to be radially diffusing throughout the arterial wall, thereby serving as the signal cueing compliance changes across the arterial wall's layers. In a study using this information, dimensionless parameters arising from the model, offering a quantitative assessment of the arterial wall's physical state, are evaluated for 12 healthy subjects, based on the 19 BAFMD responses that were obtained from them.
The present disclosure includes features derived in part from a patient study in which BAFMD processes were performed. The study was approved by the institutional review committee. BAFMD test was performed in the morning with all subjects fasting. The ultrasound scanner (ZONARE Medical Systems, Bernardo, CA, USA) was equipped with a broadband high resolution L14-5 MHz hockey stick transducer. Nineteen data sets were obtained from 12 healthy subjects aged 23-66. First, to make sure it was safe to perform the test, each subject, lying supine, had their blood pressure checked. While monitoring the brachial artery, an ischemic pressure cuff wrapped around their upper arm was inflated and maintained at 250 mmHg, as shown in. At the 5-minute mark, after the artery is completely shrunken, the cuff is suddenly deflated causing the blood flow to rush back into the artery. The recovery of the artery was monitored for about 2-5 minutes and recorded in video clips.includes an image of the artery andincludes a graph of the recovery of the artery over time.
The video clips were processed for the Diameter vs. Time data in MATLAB.illustrates how pixels inside the lumen stand out as darker than those corresponding to the surrounding tissue. This contrast in brightness was the main indicator used in the MATLAB codes that were written to extract the diameter at each frame.shows a typical FMD response for 5 minutes after the cuff is deflated. The response features an initial, relatively sharp increase in the diameter, followed by a brief dwelling phase, and then a slow recovery to its baseline.
As shown by the experimental studies cited above, sustained forward wall shear stress induces the release of vasodilators which in turn prompts the increase of the wall's compliance (the decrease of its stiffness). In the described study, considering mechanotransduction throughout the wall's thickness would require accounting for the time it takes the WSS signal picked up at the EGL to seep through the arterial wall. Since the biochemistry of vasodilators affecting smooth muscle cells is outside the scope of the proposed theory, a surrogate property, s (N/m), is visualized as diffusing radially throughout the wall's layers, delivering the “message” that these layers should soften (decrease their stiffness) accordingly. An illustration is shown in.
From its unit of stress, N/m, it can be seen that s is the equivalent shear stress that would be sensed by a certain location inside the wall if it were in contact with the blood flow, given that location's radial distance from the inner boundary. This is simply inspired from basic mass transfer principles based on which a diffusing substance's (in this case a vasodilator) concentration is attenuated as it diffuses away from the source. Since a higher value of WSS induces more vasodilator production, it can be seen why introducing this property as an alternate “messenger” is appropriate, and as will be evident shortly, theoretically convenient. A helpful analogy would be that the s surrogacy for the vasodilators' concentration, is like the temperature's surrogacy for the internal energy density. As the diffusion of s is intended to represent that of the vasodilators, the traditional first order diffusion law is assumed to govern it.
In cylindrical coordinates, s is then governed by Eq. 1:
where t(s) is time, r is the radial distance from the center, and α(m/s) is the diffusivity.
Since s is equivalent to a sensed WSS at a distance as explained in the previous subsection, the same reasoning adopted in our precursor study,-, by Sidnawi, B., Chen, Z., Sehgal, C. & Wu, Q,107, 103756 (2020), the entirety of which is incorporated by reference herein, for the wall's lumped circumferential stiffness, will be applied here for the modulus of elasticity E(N/m). The value of the local modulus of the wall, E(s), that would be reached if exposure to s is sustained (denoted by the s subscript in E) for a long period of time, is assumed to take an exponential decreasing function from a maximum value Efor s=0, to a minimum asymptotic value, E, as s→∞. This depicts the softening effect that WSS (driving the diffusion of s) has on the arterial wall. Eq. 2 below governs E.()=() (2)
β(m/N) is a characteristic property that indicates the wall's resistance to a changing value of s. a higher value means a lower resistance. This behavior is illustrated in the graph of.
The instantaneous value of the modulus, E(t), like many physiological processes, is expected to be transient. This means that after a sustained exposure to a value sof s, entailing a modulus value of E(s) (Eq. 2), if at t=t, s suddenly shifts to a new value s, the modulus would take some time to reach E(s) as illustrated in.
For t≥t, the shape of this response is also assumed to be exponential as follows:()−()=(()−()) (3)
ξ(s) is a property of the artery that quantifies its responsiveness to a changing WSS. A greater value of ξ indicates a more responsive artery. In a similar procedure to that followed in the-study, by thinking of a s(t) time signal as a series of infinitesimal steps, the equation governing E(t) in response to any function s(t) can be derived from Eqs. 1 and 2. According to Eq. 1, at any instant during a transient response E(t) the following holds: E<E(t)<E. Therefore, for any value that E(t) takes, there exists a value of s, ŝ such that:()=() (4)the incremental change in E is: dE=E(t+dt)−E(t), which from Eq. 3 becomes:()−() (5)
Applying Eq. 3 to the difference in Eq. 5, one obtains()−()=((τ)−())(1−) (6)
Hence, substituting Eq. 4 into Eq. 6,=(()−())(1−) (7)
Expanding einto a Maclaurin series and only retaining the first order term as dt→0,=(()−())ξ (8)or,
The rising-and-dwelling part of the observed FMD response for all cases is a slow process. This behavior makes it reasonable to assume that the expanding arterial wall is in a quasi-static equilibrium between the internal pressure pushing it outwards, and the opposing elastic forces resisting the deformation. When the blood flow q(m/s) is allowed back in, the WSS will be at its highest level. As the resulting diffusion of s takes place, the arterial layers will gradually soften, and the artery will therefore expand to maintain the balance between its wall's internal elastic forces and the blood pressure, decreasing the WSS in the process.
Since a purely oscillatory WSS, devoid of a forward component, has been shown to be incapable of stimulating vasodilation, only the steady components of the WSS and local pressure signals will be considered in this study. Also, with the subject lying in a restful state, and the artery being back open, allowing blood flow to the lower arm branch of the body to be reestablished, the steady component of the pressure signal at the examined location of the brachial artery will be assumed to be constant. This is because the steady component of the pressure drop from the heart's left ventricle over the fixed distance to the examined location of the brachial artery near the elbow can be reasonably assumed constant with the steady component of the blood flow rate, q, reestablished to that branch of the body.
The problem is setup as an initially undeformed cylindrical shell with an inner radius, r(m), and outer radius, r(m). A sketch is shown in the.
At t=0, an internal pressure p(N/m) with a flow rate q are imposed. As the expansion is quasi-static, the steady component of the wall shear stress, τ(N/m), would correspond to the parabolic velocity profile in a Poiseuille flow:
where μ(kg/m·s) is the blood's dynamic viscosity, and R(t) is the inner deformed radius. τ(t) is in fact s(r,t). If u(r,t) denotes the radial displacement, and u(t) denotes the displacement of the inner boundary, then R(t)=r+u(t) and Eq. 10 may be rewritten as:
where
The system of equilibrium equations governing the wall's expansion is:
where σ is the stress tensor, and θ is the orthoradial coordinate. The wall's constitutive equations are those given by the standard elasticity theory:
where E is the elasticity modulus, treated as a variable quantity as illustrated in, ν is Poisson's ratio, assumed to take on the typical value of about 0.3 in this study, and ε is the strain tensor. As the problem is axisymmetric, dependence on θ vanishes
Also, the shear stress, σwould have to assume the same value in both clockwise and counterclockwise directions, since axisymmetry renders both directions equivalent. This can only be possible if σ=0. Recognizing that
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May 26, 2026
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